Geodesic
An example of the decomposition non-uniqueness for a 3-dimensional geometric object
Čelâbinskij fiziko-matematičeskij žurnal, Tome 4 (2019) no. 3, pp. 265-275

Voir la notice de l'article provenant de la source Math-Net.Ru

In 1942, M.H.A. Newman formulated and proved a simple lemma that has been very useful in various areas of mathematics in particular in algebra and Gröbner — Shirshov bases theory. It was later called Diamond Lemma, since its key design is graphically depicted as a rhombus (diamond symbol). In 2005, I proposed a new version of this lemma, designed to solve geometric problems, and proved existence and uniqueness theorems for primary decompositions of various geometric objects: 3-dimensional manifolds, knots in thickened surfaces, knotted graphs, knotted theta curves in 3-dimensional manifolds. It turned out that all geometric objects of the mentioned types allow primary decomposition, but in some cases (for example, for orbifolds) uniqueness decomposition is absent. This article presents this new version of the lemma and an algorithm for its application. I propose a theorem that uses Diamond Lemma to prove it, and a counterexample showing the impossibility of omitting one of the conditions of the theorem.
Keywords: 3-dimensional manifold, knot, knotted graph, Diamond Lemma, prime decompositions of geometric objects. 
@article{CHFMJ_2019_4_3_a1,
     author = {S. V. Matveev},
     title = {An example of the decomposition non-uniqueness for a 3-dimensional geometric object},
     journal = {\v{C}el\^abinskij fiziko-matemati\v{c}eskij \v{z}urnal},
     pages = {265--275},
     publisher = {mathdoc},
     volume = {4},
     number = {3},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHFMJ_2019_4_3_a1/}
}
TY  - JOUR
AU  - S. V. Matveev
TI  - An example of the decomposition non-uniqueness for a 3-dimensional geometric object
JO  - Čelâbinskij fiziko-matematičeskij žurnal
PY  - 2019
SP  - 265
EP  - 275
VL  - 4
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHFMJ_2019_4_3_a1/
LA  - ru
ID  - CHFMJ_2019_4_3_a1
ER  - 
%0 Journal Article
%A S. V. Matveev
%T An example of the decomposition non-uniqueness for a 3-dimensional geometric object
%J Čelâbinskij fiziko-matematičeskij žurnal
%D 2019
%P 265-275
%V 4
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHFMJ_2019_4_3_a1/
%G ru
%F CHFMJ_2019_4_3_a1
S. V. Matveev. An example of the decomposition non-uniqueness for a 3-dimensional geometric object. Čelâbinskij fiziko-matematičeskij žurnal, Tome 4 (2019) no. 3, pp. 265-275. http://geodesic.mathdoc.fr/item/CHFMJ_2019_4_3_a1/